Abstract

As the capacity of the next-generation passive optical network (PON) is reaching 100 Gb/s and beyond, cost-effective transceivers have been widely discussed. In this work, we provide a comprehensive comparison of various simplified coherent and direct detection (DD) schemes operating at a ${100\; {\rm Gb/s/}}\lambda$ 4-ary pulse amplitude modulation signal through numerical simulation. According to the cost, the coherent receivers can be divided into three levels: intensity-only coherent receivers and phase-insensitive and phase-sensitive complex-value coherent receivers. The received power sensitivity at back-to-back, influence of laser frequency offset, local oscillator power, laser linewidth, analog-to-digital convertor resolution, fiber dispersion, and hardware complexity are investigated and analyzed for each transceiver structure. The results show the following: (1) Transmitter-side optical amplification is suggested for DD and intensity-only coherent receivers to meet the 29 dB power budget requirement, and these schemes have a large dispersion penalty. (2) Compared with ${3} \times {3}$ coupler and ${2} \times {4}$ hybrid-based coherent receivers, ${2} \times {2}$ coupler and balanced-photodiode-based heterodyne detection exhibit similar performance with a simpler structure. (3) Both phase-insensitive and phase-sensitive complex-value coherent receivers have superior power budget and dispersion tolerance, and their application in PON would depend on the cost.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (4)

2019 (2)

2018 (4)

2017 (4)

2016 (3)

2014 (1)

E. Ciaramella, “Polarization-independent receivers for low-cost coherent OOK systems,” IEEE Photon. Technol. Lett. 26, 548–551 (2014).
[Crossref]

2012 (1)

2010 (1)

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16, 1164–1179 (2010).
[Crossref]

2009 (2)

2008 (1)

2006 (1)

K. Kikuchi, “Phase-diversity homodyne detection of multilevel optical modulation with digital carrier phase estimation,” IEEE J. Sel. Top. Quantum Electron. 12, 563–570 (2006).
[Crossref]

1992 (1)

Y. H. Ja, “Analysis of four-port optical fiber ring and loop resonators using a 3 × 3 fiber coupler and degenerate two-wave mixing,” IEEE J. Quantum Electron. 28, 2749–2757 (1992).
[Crossref]

1987 (1)

B. Glance, “Polarization independent coherent optical receiver,” J. Lightwave Technol. 5, 274–276 (1987).
[Crossref]

Antonelli, C.

Aref, V.

Atzmon, Y.

Bayvel, P.

Buchali, F.

Buelow, H.

Chagnon, M.

Chandrasekhar, S.

Che, D.

Chen, X.

Chien, H.

J. Zhang, J. Yu, H. Chien, J. S. Wey, M. Kong, X. Xin, and Y. Zhang, “Demonstration of 100-Gb/s/λ PAM-4 TDM-PON supporting 29-dB power budget with 50-km reach using 10G-class O-band DML transmitters,” in Optical Fiber Communication Conference (2019), paper Th4C.3.

Chou, E.

V. Houtsma, E. Chou, and D. van Veen, “92 and 50  Gbps TDM-PON using neural network enabled receiver equalization specialized for PON,” in Optical Fiber Communication Conference (2019), paper M2B.6.

Ciaramella, E.

E. Ciaramella, “Assessment of a polarization-independent DSP-free coherent receiver for intensity-modulated signals,” J. Lightwave Technol. 38, 676–683 (2020).
[Crossref]

E. Ciaramella, “Polarization-independent receivers for low-cost coherent OOK systems,” IEEE Photon. Technol. Lett. 26, 548–551 (2014).
[Crossref]

Dischler, R.

Edvold, B.

Emmerich, R.

Engenhardt, K. M.

Erkilinç, M.

Erkilinç, M. S.

Erkilinç, S.

Fatadin, I.

Ferreira, R. M.

Forestieri, E.

Freund, R.

Galdino, L.

Gerard, T.

Gersler, T.

Glance, B.

B. Glance, “Polarization independent coherent optical receiver,” J. Lightwave Technol. 5, 274–276 (1987).
[Crossref]

Gnauck, A. H.

Guiomar, F. P.

Guo, Y.

Y. Zhu, B. Yang, Y. Zhong, Z. Liu, Y. Guo, J. S. Wey, X. Huang, and Z. Ma, “Performance comparison of coherent and direct detection schemes for 50G PON,” in Optical Fiber Communication Conference (2020), paper W1E.3.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Habel, K.

Harstead, E.

Hoffmann, S.

Houtsma, V.

Hu, Q.

Hu, W.

Huang, L.

Huang, X.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Y. Zhu, B. Yang, Y. Zhong, Z. Liu, Y. Guo, J. S. Wey, X. Huang, and Z. Ma, “Performance comparison of coherent and direct detection schemes for 50G PON,” in Optical Fiber Communication Conference (2020), paper W1E.3.

Ives, D.

Ja, Y. H.

Y. H. Ja, “Analysis of four-port optical fiber ring and loop resonators using a 3 × 3 fiber coupler and degenerate two-wave mixing,” IEEE J. Quantum Electron. 28, 2749–2757 (1992).
[Crossref]

Jungnickel, V.

Kikuchi, K.

K. Kikuchi, “Phase-diversity homodyne detection of multilevel optical modulation with digital carrier phase estimation,” IEEE J. Sel. Top. Quantum Electron. 12, 563–570 (2006).
[Crossref]

Killey, R.

Killey, R. I.

Kong, M.

J. Zhang, J. Yu, H. Chien, J. S. Wey, M. Kong, X. Xin, and Y. Zhang, “Demonstration of 100-Gb/s/λ PAM-4 TDM-PON supporting 29-dB power budget with 50-km reach using 10G-class O-band DML transmitters,” in Optical Fiber Communication Conference (2019), paper Th4C.3.

Lavery, D.

Le, S. T.

Li, H.

H. Li, M. Luo, X. Li, X. Zhang, and S. Yu, “Demonstration of polarization-insensitive coherent 56  Gb/s/λ PAM-4 PON using real-valued Alamouti coding combined with digital pilot,” in European Conference on Optical Communication (2019), paper M.1.F.4.

Li, P.

Li, X.

J. Zhang, J. Yu, X. Li, K. Wang, W. Zhou, J. Xiao, L. Zhao, X. Pan, B. Liu, and X. Xin, “200  Gbit/s/λ PDM-PAM-4 PON system based on intensity modulation and coherent detection,” J. Opt. Commun. Network. 12, A1–A8 (2020).
[Crossref]

H. Li, M. Luo, X. Li, X. Zhang, and S. Yu, “Demonstration of polarization-insensitive coherent 56  Gb/s/λ PAM-4 PON using real-valued Alamouti coding combined with digital pilot,” in European Conference on Optical Communication (2019), paper M.1.F.4.

Li, Z.

Liao, T.

Liu, B.

J. Zhang, J. Yu, X. Li, K. Wang, W. Zhou, J. Xiao, L. Zhao, X. Pan, B. Liu, and X. Xin, “200  Gbit/s/λ PDM-PAM-4 PON system based on intensity modulation and coherent detection,” J. Opt. Commun. Network. 12, A1–A8 (2020).
[Crossref]

Liu, Z.

D. Lavery, T. Gerard, S. Erkılınç, Z. Liu, L. Galdino, P. Bayvel, and R. I. Killey, “Opportunities for optical access network transceivers beyond OOK [Invited],” J. Opt. Commun. Netw. 11, A186–A195 (2019).
[Crossref]

Y. Zhu, B. Yang, Y. Zhong, Z. Liu, Y. Guo, J. S. Wey, X. Huang, and Z. Ma, “Performance comparison of coherent and direct detection schemes for 50G PON,” in Optical Fiber Communication Conference (2020), paper W1E.3.

Livshitz, B.

Luis, R. S.

Luo, M.

H. Li, M. Luo, X. Li, X. Zhang, and S. Yu, “Demonstration of polarization-insensitive coherent 56  Gb/s/λ PAM-4 PON using real-valued Alamouti coding combined with digital pilot,” in European Conference on Optical Communication (2019), paper M.1.F.4.

Ma, Z.

Y. Zhu, B. Yang, Y. Zhong, Z. Liu, Y. Guo, J. S. Wey, X. Huang, and Z. Ma, “Performance comparison of coherent and direct detection schemes for 50G PON,” in Optical Fiber Communication Conference (2020), paper W1E.3.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Mecozzi, A.

Nazarathy, M.

Noé, R.

Pan, X.

J. Zhang, J. Yu, X. Li, K. Wang, W. Zhou, J. Xiao, L. Zhao, X. Pan, B. Liu, and X. Xin, “200  Gbit/s/λ PDM-PAM-4 PON system based on intensity modulation and coherent detection,” J. Opt. Commun. Network. 12, A1–A8 (2020).
[Crossref]

Pfau, T.

Raybon, G.

Reis, J. D.

Ruan, X.

Y. Zhu, K. Zou, X. Ruan, and F. Zhang, “Single carrier 400G transmission with single-ended heterodyne detection,” IEEE Photon. Technol. Lett. 29, 1788–1791 (2017).
[Crossref]

Savory, S. J.

Schmidt-Langhorst, C.

Schubert, C.

Schuh, K.

Secondini, M.

Shahpari, A.

Shi, J.

J. Zhang, J. S. Wey, J. Shi, and J. Yu, “Single-wavelength 100-Gb/s PAM-4 TDM-PON achieving over 32-dB power budget using simplified and phase insensitive coherent detection,” in European Conference on Optical Communication (2018), paper Tu1B.1.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Shi, K.

Shieh, W.

Shtaif, M.

Sillekens, E.

Teixeira, A. L.

Thomsen, B.

Thomsen, B. C.

Tu, Z.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

van Veen, D.

Vujicic, Z.

Wang, K.

J. Zhang, J. Yu, X. Li, K. Wang, W. Zhou, J. Xiao, L. Zhao, X. Pan, B. Liu, and X. Xin, “200  Gbit/s/λ PDM-PAM-4 PON system based on intensity modulation and coherent detection,” J. Opt. Commun. Network. 12, A1–A8 (2020).
[Crossref]

Wey, J. S.

J. S. Wey, “The outlook for PON standardization: a tutorial,” J. Lightwave Technol. 38, 31–42 (2020).
[Crossref]

J. Zhang, J. Yu, H. Chien, J. S. Wey, M. Kong, X. Xin, and Y. Zhang, “Demonstration of 100-Gb/s/λ PAM-4 TDM-PON supporting 29-dB power budget with 50-km reach using 10G-class O-band DML transmitters,” in Optical Fiber Communication Conference (2019), paper Th4C.3.

J. Zhang, J. S. Wey, J. Shi, and J. Yu, “Single-wavelength 100-Gb/s PAM-4 TDM-PON achieving over 32-dB power budget using simplified and phase insensitive coherent detection,” in European Conference on Optical Communication (2018), paper Tu1B.1.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Y. Zhu, B. Yang, Y. Zhong, Z. Liu, Y. Guo, J. S. Wey, X. Huang, and Z. Ma, “Performance comparison of coherent and direct detection schemes for 50G PON,” in Optical Fiber Communication Conference (2020), paper W1E.3.

Winzer, P.

Winzer, P. J.

Xiao, J.

J. Zhang, J. Yu, X. Li, K. Wang, W. Zhou, J. Xiao, L. Zhao, X. Pan, B. Liu, and X. Xin, “200  Gbit/s/λ PDM-PAM-4 PON system based on intensity modulation and coherent detection,” J. Opt. Commun. Network. 12, A1–A8 (2020).
[Crossref]

Xie, C.

Xin, X.

J. Zhang, J. Yu, X. Li, K. Wang, W. Zhou, J. Xiao, L. Zhao, X. Pan, B. Liu, and X. Xin, “200  Gbit/s/λ PDM-PAM-4 PON system based on intensity modulation and coherent detection,” J. Opt. Commun. Network. 12, A1–A8 (2020).
[Crossref]

J. Zhang, J. Yu, H. Chien, J. S. Wey, M. Kong, X. Xin, and Y. Zhang, “Demonstration of 100-Gb/s/λ PAM-4 TDM-PON supporting 29-dB power budget with 50-km reach using 10G-class O-band DML transmitters,” in Optical Fiber Communication Conference (2019), paper Th4C.3.

Xue, L.

Yang, B.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Y. Zhu, B. Yang, Y. Zhong, Z. Liu, Y. Guo, J. S. Wey, X. Huang, and Z. Ma, “Performance comparison of coherent and direct detection schemes for 50G PON,” in Optical Fiber Communication Conference (2020), paper W1E.3.

Yang, W.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Yi, L.

Yu, J.

J. Zhang, J. Yu, X. Li, K. Wang, W. Zhou, J. Xiao, L. Zhao, X. Pan, B. Liu, and X. Xin, “200  Gbit/s/λ PDM-PAM-4 PON system based on intensity modulation and coherent detection,” J. Opt. Commun. Network. 12, A1–A8 (2020).
[Crossref]

J. Zhang, J. S. Wey, J. Shi, and J. Yu, “Single-wavelength 100-Gb/s PAM-4 TDM-PON achieving over 32-dB power budget using simplified and phase insensitive coherent detection,” in European Conference on Optical Communication (2018), paper Tu1B.1.

J. Zhang, J. Yu, H. Chien, J. S. Wey, M. Kong, X. Xin, and Y. Zhang, “Demonstration of 100-Gb/s/λ PAM-4 TDM-PON supporting 29-dB power budget with 50-km reach using 10G-class O-band DML transmitters,” in Optical Fiber Communication Conference (2019), paper Th4C.3.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Yu, S.

H. Li, M. Luo, X. Li, X. Zhang, and S. Yu, “Demonstration of polarization-insensitive coherent 56  Gb/s/λ PAM-4 PON using real-valued Alamouti coding combined with digital pilot,” in European Conference on Optical Communication (2019), paper M.1.F.4.

Zhang, F.

Y. Zhu, K. Zou, X. Ruan, and F. Zhang, “Single carrier 400G transmission with single-ended heterodyne detection,” IEEE Photon. Technol. Lett. 29, 1788–1791 (2017).
[Crossref]

Zhang, J.

J. Zhang, J. Yu, X. Li, K. Wang, W. Zhou, J. Xiao, L. Zhao, X. Pan, B. Liu, and X. Xin, “200  Gbit/s/λ PDM-PAM-4 PON system based on intensity modulation and coherent detection,” J. Opt. Commun. Network. 12, A1–A8 (2020).
[Crossref]

J. Zhang, J. S. Wey, J. Shi, and J. Yu, “Single-wavelength 100-Gb/s PAM-4 TDM-PON achieving over 32-dB power budget using simplified and phase insensitive coherent detection,” in European Conference on Optical Communication (2018), paper Tu1B.1.

J. Zhang, J. Yu, H. Chien, J. S. Wey, M. Kong, X. Xin, and Y. Zhang, “Demonstration of 100-Gb/s/λ PAM-4 TDM-PON supporting 29-dB power budget with 50-km reach using 10G-class O-band DML transmitters,” in Optical Fiber Communication Conference (2019), paper Th4C.3.

J. Zhang, J. S. Wey, J. Shi, J. Yu, Z. Tu, B. Yang, W. Yang, Y. Guo, X. Huang, and Z. Ma, “Experimental demonstration of unequally spaced PAM-4 signal to improve receiver sensitivity for 50-Gbps PON with power-dependent noise distribution,” in Optical Fiber Communication Conference (2018), paper M2B.3.

Zhang, X.

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V. Houtsma, E. Chou, and D. van Veen, “92 and 50  Gbps TDM-PON using neural network enabled receiver equalization specialized for PON,” in Optical Fiber Communication Conference (2019), paper M2B.6.

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Figures (11)

Fig. 1.
Fig. 1. Transmitter and receiver structures of DD and coherent detection schemes. Tx, transmitter; Rx, receiver; DAC, digital-to-analog convertor; ADC, analog-to-digital convertor; EML, external modulated laser; EAM, electro-absorption modulator; MZM, Mach-Zehnder modulator; DP, dual-polarization; SOA, semiconductor optical amplifier; PIN, positive-intrinsic-negative; SPD, single-ended photodiode; BPD, balanced photodiode; LPF, low-pass filter; Het., heterodyne; Int., intradyne.
Fig. 2.
Fig. 2. (a) Transmitter-side and (b) receiver-side DSP stack. DD, direct detection; Rx, receiver; Norm., normalization; KK, Kramers–Kronig; FOE, frequency offset estimation.
Fig. 3.
Fig. 3. Typical eye diagrams and constellations of simplified coherent detection and DD schemes at different received optical power, respectively. ROP, received optical power; Het., heterodyne; Int., intradyne; Cx., complex-value; Eq., equalization; PR, phase recovery.
Fig. 4.
Fig. 4. Received power spectral density of (a) ${2} \times {2}$ BPD-based heterodyne and (b) ${2} \times {4}$ DP intradyne coherent receivers with phase-sensitive complex-value reception, respectively. w/, with.
Fig. 5.
Fig. 5. Simulated receiver power sensitivity of DD, intensity-only, phase-insensitive, and phase-sensitive coherent detection schemes at back-to-back scenario. ROP, received optical power; In., intensity-only; PI, phase-insensitive; PS, phase-sensitive; Cx., complex-value.
Fig. 6.
Fig. 6. Simulated sensitivity penalty versus frequency offset for different intensity-only coherent receivers at BTB, respectively.
Fig. 7.
Fig. 7. Simulated sensitivity penalty versus LO power for intensity-only, phase-insensitive, and phase-sensitive coherent detection schemes at BTB, respectively.
Fig. 8.
Fig. 8. Simulated sensitivity penalty versus frequency offset for different coherent receivers at BTB.
Fig. 9.
Fig. 9. Simulated sensitivity penalty of ADC resolution for DD, intensity-only, phase-insensitive, and phase-sensitive coherent receivers, respectively.
Fig. 10.
Fig. 10. Simulated fiber dispersion penalty with FFE: (a) ideal ADC, (b) 8 bit ADC, (c) 6 bit ADC, and (d) 4 bit ADC; and with ${\rm FFE} + {\rm DFE}$: (e) ideal ADC, (f) 8 bit ADC, (g) 6 bit ADC, and (h) 4 bit ADC, respectively. w/, with.
Fig. 11.
Fig. 11. Hardware complexity and sensitivity of DD: intensity-only, phase-insensitive, and phase-sensitive coherent receivers, respectively. Mod., modulator; OA, optical amplifier; OH, optical hybrid; AC, analog process circuit; In., intensity-only; PI, phase-insensitive; PS, phase-sensitive.

Tables (4)

Tables Icon

Table 1. Simulation Parameters

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Table 2. Receiver Sensitivity at B E R = 1 × 10 2

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Table 3. Penalty at B E R = 1 × 10 2 with 3 dBm LO Power

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Table 4. Penalty at B E R = 1 × 10 2 with 60 ps/nm Fiber Dispersion and an Ideal ADC

Equations (9)

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I x / y | L + E x / y 2 | 2 = | L | 2 + 2 Re { E x / y L } + | E x / y | 2 2 | L | 2 + 2 Re { E x / y L } 2 = 2 | E x / y | | L | cos ( Δ ω t + Δ φ ) + | L | 2 2 .
r = I x 2 + I y 2 = 1 + cos ( 2 Δ ω t + 2 Δ φ ) 2 ( | E x | 2 + | E y | 2 ) | L | 2 = 1 + cos ( 2 Δ ω t + 2 Δ φ ) 2 | L | 2 S .
φ x / y ( t ) = H [ ln ( I x / y ) ] , E x / y = I x / y exp [ j φ x / y ( t ) ] .
I x / y | L + E x / y 2 | 2 | L E x / y 2 | 2 = 2 Re { E x / y L } = 2 | E x / y | | L | cos ( Δ ω t + Δ φ ) .
( a b b b a b b b a ) ( E x E y 0 L L 0 ) = ( a E x + b L a E y + b L b E x + b L b E y + a L b E x + a L b E y + b L ) , a = 2 exp ( j 2 π / 9 ) / 3 + exp ( j 4 π / 9 ) / 3 , b = exp ( j 4 π / 9 ) / 3 exp ( j 2 π / 9 ) / 3.
r = ( I 1 I 2 ) 2 + ( I 2 I 3 ) 2 + ( I 3 I 1 ) 2 2 3 ( 1 sin ( 2 θ ) sin ( π 6 2 Δ ω t Δ φ ) ) | L | 2 S .
I x 1 / y 1 | a E x / y + b L | 2 [ | E x / y | 2 + 2 | E x / y | | L | cos ( Δ φ + 2 3 π ) + | L | 2 ] / 3 , I x 2 / y 2 | b E x / y + b L | 2 [ | E x / y | 2 + 2 | E x / y | | L | cos ( Δ φ ) + | L | 2 ] / 3 , I x 3 / y 3 | b E x / y + a L | 2 [ | E x / y | 2 + 2 | E x / y | | L | cos ( Δ φ 2 3 π ) + | L | 2 ] / 3 .
r x I / y I = I x 2 / y 2 0.5 ( I x 1 / y 1 + I x 3 / y 3 ) = | E x / y | | L | cos Δ φ , r x Q / y Q = 3 2 ( I x 1 / y 1 I x 3 / y 3 ) = | E x / y | | L | sin Δ φ , r = r xI 2 + r xQ 2 + r yI 2 + r yQ 2 = | L | 2 S .
I x I / y I | E x / y + L 2 | 2 | E x / y L 2 | 2 = R e { E x / y L } , I x Q / y Q | E x / y + j L 2 | 2 | E x / y j L 2 | 2 = I m { E x / y L } , r = r xI 2 + r xQ 2 + r yI 2 + r yQ 2 = | L | 2 S .